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Approximate zero modes for the Pauli operator on a region

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Approximate zero modes for the Pauli operator on a region. / Elton, Daniel.

In: Journal of Spectral Theory, Vol. 6, No. 2, 2016, p. 373-413.

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Elton, D 2016, 'Approximate zero modes for the Pauli operator on a region', Journal of Spectral Theory, vol. 6, no. 2, pp. 373-413. https://doi.org/10.4171/JST/127

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Elton, Daniel. / Approximate zero modes for the Pauli operator on a region. In: Journal of Spectral Theory. 2016 ; Vol. 6, No. 2. pp. 373-413.

Bibtex

@article{11f4508908d24e69bec40944f5ea2557,
title = "Approximate zero modes for the Pauli operator on a region",
abstract = "Let $\Pauli{\reg,t\magp}$ denoted the Pauli operator on a bounded open region $\reg\subset\R^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$.Assume that the corresponding magnetic field $B=\mathrm{Curl}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula\[\mathsf{N}_{\Omega,tA}(\lambda(t))\,=\,\frac{t}{2\pi}\int_{\Omega}\lvert{B(x)}\rvert\,d x\;+o(t)\]as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$.The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.",
keywords = "Pauli operator, eigenvalue asymptotics, approximate zero modes",
author = "Daniel Elton",
year = "2016",
doi = "10.4171/JST/127",
language = "English",
volume = "6",
pages = "373--413",
journal = "Journal of Spectral Theory",
issn = "1664-039X",
publisher = "European Mathematical Society Publishing House",
number = "2",

}

RIS

TY - JOUR

T1 - Approximate zero modes for the Pauli operator on a region

AU - Elton, Daniel

PY - 2016

Y1 - 2016

N2 - Let $\Pauli{\reg,t\magp}$ denoted the Pauli operator on a bounded open region $\reg\subset\R^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$.Assume that the corresponding magnetic field $B=\mathrm{Curl}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula\[\mathsf{N}_{\Omega,tA}(\lambda(t))\,=\,\frac{t}{2\pi}\int_{\Omega}\lvert{B(x)}\rvert\,d x\;+o(t)\]as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$.The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.

AB - Let $\Pauli{\reg,t\magp}$ denoted the Pauli operator on a bounded open region $\reg\subset\R^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$.Assume that the corresponding magnetic field $B=\mathrm{Curl}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula\[\mathsf{N}_{\Omega,tA}(\lambda(t))\,=\,\frac{t}{2\pi}\int_{\Omega}\lvert{B(x)}\rvert\,d x\;+o(t)\]as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$.The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.

KW - Pauli operator

KW - eigenvalue asymptotics

KW - approximate zero modes

U2 - 10.4171/JST/127

DO - 10.4171/JST/127

M3 - Journal article

VL - 6

SP - 373

EP - 413

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

SN - 1664-039X

IS - 2

ER -